Chapter 13 Magnetically Coupled Circuits EEEB123 Circuit Analysis 2 Chapter 13 Magnetically Coupled Circuits Materials from Fundamentals of Electric Circuits (4th Edition), Alexander & Sadiku, McGraw-Hill Companies, Inc.

Magnetically Coupled Circuit Chapter 13 13.1 Introduction -What is a transformer? 13.2 Mutual Inductance 13.3 Energy in a Coupled Circuit 13.5 Ideal Transformers

13.1 What is a transformer? (1) It is an electrical device designed on the basis of the concept of magnetic coupling. It uses magnetically coupled coils to transfer energy from one circuit to another. It is the key circuit elements for stepping up or stepping down ac voltages or currents, impedance matching, isolation, etc.

13.2 Mutual Inductance (1) It is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H). Open circuit Mutual inductance (or induced voltage) across coil 2. Open circuit Mutual inductance (or induced voltage) across coil 1. M21 = Mutual inductance of coil 2 with respect to coil 1. M12 = Mutual inductance of coil 1 with respect to coil 2.

13.2 Mutual Inductance (2) If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil. Illustration of the dot convention.

13.2 Mutual Inductance (3) Dot convention

13.2 Mutual Inductance (4) Dot convention for coils in series; the sign indicates the polarity of the mutual voltage; (a) series-aiding connection, (b) series-opposing connection.

13.2 Mutual Inductance (5) Time-domain analysis of a circuit containing coupled coils. Frequency-domain analysis of a circuit containing coupled coils.

13.2 Mutual Inductance (6) Example 13.1 Calculate the phasor currents I1 and I2 in the circuit shown below. Ans: *Refer to textbook pp. 561-562

13.3 Energy in a Coupled Circuit (1) The coupling coefficient, k, is a measure of the magnetic coupling between two coils; 0≤k≤1. The instantaneous energy stored in the circuit is given by Positive sign for mutual term if both currents enter or leave the dotted terminals. Negative sign if otherwise.

13.3 Energy in a Coupled Circuit (2) Example 13.3 Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductors at time t = 1s if v=60cos(4t +30°) V. Ans: k=0.56; w(1)=20.73J Refer to textbook pp. 566-567

13.5 Ideal Transformer (1) A transformer is said to be ideal if: Coils have very large reactances (L1, L2, M ) Coupling coefficient is equal to unity (k = 1) Primary and secondary coil are loss less (R1 = 0 = R2) An ideal transformer is a unity-coupled, lossless transformer in which the primary and secondary coils have infinite self-inductances.

V2>V1→ step-up transformer V2<V1→ step-down transformer 13.5 Ideal Transformer (2) V2>V1→ step-up transformer V2<V1→ step-down transformer Ideal Transformer Circuit symbol

13.5 Ideal Transformer (3) Polarity of V and direction of I: If V1 and V2 are both positive or both negative at the dotted terminals, use +n in . Otherwise use –n . If I1 and I2 both enter or both leave the dotted terminals, use –n in . Otherwise use +n.

13.5 Ideal Transformer (4)

13.5 Ideal Transformer (5) V1 can be expressed in terms of V2 and I1 in terms of I2, or vice versa: or Complex power in primary winding is: showing S supplied to primary is delivered to secondary without loss.

13.5 Ideal Transformer (6) Input impedance as seen by the source figure above is: Since , we get

13.5 Ideal Transformer (7) Input impedance is also known as reflected impedance, since it appears as if load impedance is reflected to primary side. Common practice in analyzing circuit with ideal transformer is to eliminate transformer by reflecting impedances and sources from one side of the transformer to the other.

13.5 Ideal Transformer (8) Example: Want to reflect secondary side of circuit below to primary side: Find Thevenin equivalent of circuit to the right of terminals a-b. (i.e. Obtain VTh as open-circuit voltage at terminals a-b). Get ZTh by removing voltage source at secondary winding and insert a unit source at terminals a-b.

13.5 Ideal Transformer (8) Obtain VTh

13.5 Ideal Transformer (8) Obtain ZTh where

13.5 Ideal Transformer (11) The general rule for eliminating the transformer and reflecting the secondary circuit to the primary side is: divide the secondary impedance by n2, divide the secondary voltage by n, and multiply the secondary current by n.

13.5 Ideal Transformer (12) The general rule for eliminating the transformer and reflecting the primary circuit to the secondary side is: multiply the primary impedance by n2, multiply the primary voltage by n, and divide the primary current by n.

13.5 Ideal Transformer (13) Example 13.7 An ideal transformer is rated at 2400/120V, 9.6 kVA, and has 50 turns on the secondary side. Calculate: the turns ratio, the number of turns on the primary side, and the current ratings for the primary and secondary windings. Ans: This is a step-down transformer, n=0.05 N1 = 1000 turns I1 = 4A and I2 = 80A Refer to textbook pp. 578